Spectral properties of second-order, multi-point, p-Laplacian boundary value problems

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Abstract

We consider the multi-point boundary value problem - f{symbol}p (u')' = ? f{symbol}p (u), on (- 1, 1),u (± 1) = underover(?, i = 1, m±) ai± u (?i±), where p > 1, f{symbol}p (s) {colon equals} | s |p - 1 sgn s for s ? R, ? ? R, m± = 1 are integers, ?i± ? (- 1, 1), 1 = i = m±, and the coefficients ai± satisfy underover(?, i = 1, m±) | ai± | < 1 . A number ? ? R is said to be an eigenvalue of the above problem if there exists a non-trivial solution u. The spectrum is the set of eigenvalues. In this paper we obtain some basic spectral and degree-theoretic properties of this eigenvalue problem. These results have numerous applications to more general problems. As an example, a Rabinowitz-type, global bifurcation theorem is briefly described. © 2010 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)4244-4253
Number of pages10
JournalNonlinear Analysis: Theory, Methods and Applications
Volume72
Issue number11
DOIs
Publication statusPublished - 1 Jun 2010

Keywords

  • Eigenvalues
  • Multi point boundary value problem
  • p-Laplacian
  • Topological degree theory

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